A Swan length theorem and a Fong dimension theorem for Mackey algebras

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www.elsevier.com/locate/jalgebra

A Swan length theorem and a Fong dimension theorem

for Mackey algebras

Ergün Yaraneri

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

Received 3 January 2006 Available online 29 December 2006

Communicated by Michel Broué

Abstract

We first present some results about Mackey algebras of p-groups over fields of characteristic p, including their primitive idempotents and decompositions of their simple and principal indecomposable modules un-der restriction. We then apply these results together with a Green’s indecomposability theorem for Mackey algebras to obtain Mackey algebra versions of some classical results of group algebras which are mostly related to restriction, induction and dimensions of modules. Our results about dimensions include Mackey algebra analogues of Dickson’s theorem, Swan’s theorem and Fong’s dimension formula.

Keywords: Mackey functor; Dimension; Fong’s dimension formula

1. Introduction

Swan’s theorem on the composition length of a module and Fong’s theorem on the dimension are two classic applications of Clifford theory of group algebras. The theory of Mackey functors is rather different from the theory of group algebras. For instance, restriction does not preserve dimension. Nevertheless, we shall be using a Clifford theory of Mackey functors to obtain ana-logues of Swan’s theorem and Fong’s dimension theorem.

Mackey functors were introduced by J.A. Green [6] and A. Dress [5] to axiomatize repre-sentation theory of finite groups, unifying several notions like reprerepre-sentation rings, G-algebras and cohomology. Besides the definitions of Mackey functors given in [5,6], there is another one

E-mail address: yaraneri@fen.bilkent.edu.tr.

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introduced by J. Thévenaz and P. Webb in [12] which identifies Mackey functors of a finite group G over a commutative unital ring R with modules of an R-free R-algebra μR(G), called

the Mackey algebra of G over R, allowing one to adopt many module theoretical constructions. J. Thévenaz and P. Webb in [11] constructed the simple Mackey functors explicitly. We will use these comprehensive references [11,12] frequently.

As usual in Section 2, we have collected some crucial notions about Mackey functors. The structure of Mackey algebra μ_F(G)of a p-group G over an algebraically closed fieldF of charac-teristic p is studied in Section 3. By using [12], we observe thatH_GtHH= 1 is an orthogonal

decomposition of the unity of μ_F(G)as a sum of primitive idempotents, and the principal in-decomposable μ_F(G)-module μ_F(G)t_HH is the projective cover of S_H,G_F. We also observe that the center of μ_F(G)is a local algebra. Then we provide explicit decompositions of simple and principal indecomposable μ_F(G)-modules under restriction.

In later sections we use a Mackey functor version of Green’s indecomposability theorem to prove some results on lengths and dimensions of modules, including Mackey functor ver-sions of some theorems of Dickson, Swan, Dade and Fong. For many of them we need some extra assumptions on group G to guarantee that μ_F(G)-modules satisfy analogues of the the-orems mentioned above. These additional assumptions are needed because of some structural differences between Mackey algebras and group algebras. For a proper subgroup H of G, the subalgebra μ_F(H )of μ_F(G)is not unital, principal indecomposable μ_F(G)-modules may have vertices different than 1, and indecomposable μ_F(G)-modules may have vertices which are not

p-groups. These are among the main differences which prevent us from finding exact analogues of the results mentioned above.

Let P be a p-subgroup of G and S be a Sylow p-subgroup of G containing P , and let V be a finitely generated P -projectiveFG-module. Then it is known that |S : P | divides dim_FV. In particular, if V is principal indecomposable, equivalently P= 1, then the order of a Sylow

p-subgroup of G divides dim_FV. This is a theorem of Dickson. In Section 4 we provide Mackey algebra versions of these results. Let T be a finitely generated indecomposable μ_F(G)-module. We first show that dim_F↓G_PT is divisible by dim_Fμ_F(P )t₁1if T is 1-projective, which is remi-niscent of Dickson’s theorem. Also it is shown that|S : P | divides dim_F↓G_PT if P is normal and

T is P -projective.

Suppose that N is a normal subgroup of G with G/N is p-solvable. If U is a simple FG-module then by a theorem of Swan the composition length of↓G_NU divides|G : N|. We prove in Section 5 that a similar result holds for Mackey algebras. Namely, if M is a simple μ_F(G) -module with↓G_NM= 0 then the composition length of ↓G_NMdivides|G : N|. In Section 6, we provide a Fong dimension theorem for Mackey algebras.

In Section 7 we prove a result for Mackey algebras of direct products of groups which has some applications for Mackey algebras of nilpotent groups. For instance, if G is nilpotent and H is a subgroup of G such that the Sylow p-subgroup of H is not normal in the Sylow p-subgroup of G, then the dimension of every simple μ_F(G)-module of the form S_H,VG is divisible by p.

Throughout the paper, G denotes a finite group, R denotes a commutative unital ring, K denotes a field, and F denotes an algebraically closed field of characteristic p > 0. We write

H G to indicate that H is a subgroup of G. Let H G K. The notation H =GKmeans

that K is G-conjugate to H and HGKmeans that H is G-conjugate to a subgroup of K. By

the notation gH⊆ G we mean that g ranges over a complete set of representatives of left cosets of H in G, and by H gK⊆ G we mean that g ranges over a complete set of representatives of double cosets of (H, K) in G. Also we put NG(H )= NG(H )/H,gH= gHg−1, Hg= g−1H g

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and Soc(U ) the Jacobson radical and the socle of U , respectively. We will mainly work over a fieldF which is algebraically closed and of characteristic p > 0.

2. Preliminaries

In this section, we briefly summarize some crucial material on Mackey functors. For the proofs, see Thévenaz–Webb [11,12]. Recall that a Mackey functor for G over R is such that, for each subgroup H of G, there is an R-module M(H ); for each pair H, K G with

H K, there are R-module homomorphisms r_HK: M(K)→ M(H ) called the restriction map and t_HK: M(H )→ M(K) called the transfer map or the trace map; for each g ∈ G, there is an

R-module homomorphism cg_H: M(H )→ M(gH ) called the conjugation map. The following axioms must be satisfied for any g, h∈ G and H, K, L G [1,6,11,12].

(M1) If H K L, r_HL= r_HKr_KLand t_HL= t_KLt_HK; moreover r_HH = t_HH= idM(H ). (M2) c_Kgh= cgh_Kc h K. (M3) If h∈ H , c_Hh : M(H )→ M(H ) is the identity. (M4) If H K, c_Hgr_HK= r g_K g_Hc g Kand c g KtHK= t g_K g_Hc g H. (M5) (Mackey Axiom) If H L K, rHLtKL= H gK⊆LtHH∩g_Kr g_K H∩g_Kc g K.

Another possible definition of Mackey functors for G over R uses the Mackey algebra μR(G)

[1,12]: μ_Z(G) is the algebra generated by the elements r_HK, t_HK, and c_Hg, where H and K are subgroups of G such that H K, and g ∈ G, with the relations (M1)–(M7).

(M6) _H_Gt_HH =_H_Gr_HH = 1μ_Z(G).

(M7) Any other product of rHK, tHKand c g

His zero.

A Mackey functor M for G, defined in the first sense, gives a left module Mof the associative

R-algebra μR(G)= R ⊗ZμZ(G)defined by M=

HGM(H ). Conversely, if Mis a μR(G)

-module then Mcorresponds to a Mackey functor M in the first sense, defined by M(H )= t_HHM, the maps t_HK, r_HK, and c_Hg being defined as the corresponding elements of the μR(G). Moreover,

homomorphisms and subfunctors of Mackey functors for G are μR(G)-module homomorphisms

and μR(G)-submodules, and conversely.

Theorem 2.1. (See [12].) Letting H and Krun over all subgroups of G, letting g run over

representatives of the double cosets H gK⊆ G, and letting J runs over representatives of the conjugacy classes of subgroups of Hg∩ K, then tgH_Jc

g Jr

K

J comprise, without repetition, a free

R-basis of μR(G).

Given a simple Mackey functor M for G over R, there is a unique, up to G-conjugacy, sub-group H of G, called a minimal subsub-group of M, such that M(H ) is nonzero. Moreover, for such an H the RNG(H )-module M(H ) is simple, see [11].

Theorem 2.2. (See [11].) Given a subgroup H G and a simple RNG(H )-module V , then

there exists a simple Mackey functor SG_H,V for G, unique up to isomorphism, such that H is a minimal subgroup of S_H,VG and S_H,VG (H ) ∼= V . Moreover, up to isomorphism, every simple

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Mackey functor for G has the form S_H,VG for some H G and simple RNG(H )-module V . Two

simple Mackey functors S_H,VG and S_HG,V are isomorphic if and only if, for some element g∈ G,

we have H=gH and V∼= c_Hg(V ).

We now recall the definitions of restriction, induction and conjugation for Mackey functors [1,9,11,12]. Let M and T be Mackey functors for G and H , respectively, where H G, then the restricted Mackey functor↓G_HMis the μR(H )-module 1μR(H )M and the induced Mackey

functor↑G_HT is the μR(G)-module μR(G)1μR(H )⊗μR(H )T, where 1μR(H )denotes the unity of

μR(H ). For g∈ G, the conjugate Mackey functor |g_HT =gT is the μR(gH )-module T with the

module structure given for any x∈ μR(gH )and t∈ T by x.t = (γg−1xγg)twhere γgis the sum

of all cg_X with X ranging over subgroups of H . Obviously, one has |g_LS_H,VL ∼= S_ggL

H,cg_H(V ). The

subgroup{g ∈ NG(H ): gT T } of NG(H )is called the inertia group of T in NG(H ).

Given H G K and a Mackey functor M for K over R, the following is the Mackey decomposition formula for Mackey algebras [12], which will be of great use,

↓L H↑ L KM ∼= H gK⊆L ↑H H∩g_K↓ g_K H∩g_K|g_KM.

We let B_RGdenote Burnside functor for G, see [12].

Theorem 2.3. (See [12].)

(i) μR(G)tHH ↑GHBRH.

(ii) B_FGis indecomposable if and only if G is a p-group.

(iii) Let G be a p-group. Then S_H,G_F(K) is nonzero if and only if H=GK.

As a last result in this section, we record a Mackey algebra version of Green’s indecompos-ability theorem which will be used frequently.

Theorem 2.4. (See [13].) Let N be a normal subgroup of G. Let S be a finitely generated

inde-composable Mackey functor for N overF, and let L be the inertia group of S. Then, ↑G_NS is an indecomposable Mackey functor for G overF if and only if L/N is a p-group.

3. Mackey functors ofp-groups

In this section, using some results from [12], we find an orthogonal primitive idempotent decomposition of the unity of Mackey algebra of a p-group overF, an algebraically closed field of characteristic p. As in [12] we let P_H,VG denote the projective cover of the simple Mackey functor S_H,VG for G.

Proposition 3.1. Let G be a p-group. Then for any subgroup H of G,

(i) The idempotent t_HH ∈ μ_F(G) is primitive.

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Proof. (i) The idempotent t_HH is primitive if and only if μ_F(G)t_HH is an indecomposable μ_F(G) -module. Indeed, μ_F(G)t_HH ↑G_HB_FH is indecomposable by 2.3 and 2.4.

(ii) As_H_Gt_HH = 1 is an expression of 1 as a sum of primitive orthogonal idempotents,

the projective cover P_H,G_Fof the simple μ_F(G)-module S_H,G_F must be isomorphic to μ_F(G)t_XX

for some subgroup X of G. Therefore the simple modules μ_F(G)t_XX/J (μ_F(G))t_XX and S_H,G_F must be isomorphic, implying that t_XXdoes not annihilate S_H,G_F, that is S_H,G_F(X)= 0. By 2.3(iii) X=GH. 2

The previous result can be proved without using 2.4. Indeed, it follows by 2.3 and [12, 8.6], a result which express↑G_HB_KH explicitly as a direct sum of principal indecomposable μ_K(G) -modules, for large enough fieldsK.

We record an immediate consequence of 3.1.

Corollary 3.2. Let G be a p-group. Then for any subgroups H and K of G,

(i) μ_F(G)t_HH μ_F(G)t_KK if and only if H=GK.

(ii) If H K then ↑G_KP_H,K_F P_H,G_F.

Proof. (i) Obvious by 3.1(ii).

(ii) Using 3.1(ii),↑G_LP_H,L_F ↑G_L↑L_HB_FH ↑G_HB_FH P_H,G_F. 2

The group algebraFG is local if G is a p-group. For Mackey algebras we have

Proposition 3.3.

(i) If G is a p-group, then the center of μ_F(G) is a local algebra, so1 is a block idempotent of

μ_F(G).

(ii) If the center of μ_K(G) is a local algebra then K is of characteristic p > 0 and G is a p-group.

Proof. (i) For any subgroups H and K of G, the primitive idempotents t_HH and t_KK of μF(G)lie in the same block because t_HHμ_F(G)t_KK is nonzero (it contains t₁Hr₁K). As_H_Gt_HH= 1 is an

orthogonal decomposition, 1 must be a block idempotent.

(ii) The Burnside algebra B_K(G) embeds into the center of μ_K(G), see [12]. Hence 1∈

B_K(G)is a primitive idempotent and the result follows by [4]. 2

For a p-group G, the radical J (FG) of FG is the kernel of the augmentation map. We see now that a similar result holds for Mackey algebras. Consider a map μR(G)→ R whose image

at the basis element t_GGis 1 and at the other basis elements tgH_Jc g

JrJK are 0, see 2.1 for the basis

elements. It is a routine checking that the above map is an R-algebra epimorphism. We let ψ denote its restriction to the center Z(μR(G))of μR(G). If G is a p-group then as Z(μ_F(G))is

local it is clear that the radical of Z(μF(G))is equal to the kernel of ψ , which is also equal to the set of all elements of Z(μ_F(G))annihilating the simple μ_F(G)-module S_G,G_F.

For future use we next state a Mackey algebra version of Nakayama’s reciprocities for group algebras whose first three parts depend on the adjointness of restriction and induction functors and whose last part depend on the symmetricity of group algebras. For Mackey algebras, we

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have the same adjointness properties of restriction and induction functors, see [11]. So a slight modification of the proof of the result for group algebras implies

Proposition 3.4. LetK be algebraically closed and H be a subgroup of G. Let S be a simple

μ_K(G)-module and T a simple μ_K(H )-module, and let P (S) and P (T ) denote their projective covers. Then,

(i) The multiplicity of S as a simple constituent of↑G_HT /J (↑G_HT ) is equal to the multiplicity of T as a simple constituent of Soc(↓G_HS).

(ii) The multiplicity of P (T ) as a direct summand of↓G_HP (S) is equal to the multiplicity of S as a simple constituent of↑G_HT .

(iii) The multiplicity of P (S) as a direct summand of↑G_HP (T ) is equal to the multiplicity of T as a simple constituent of↓G_HS.

(iv) If μ_K(H ) is a symmetric algebra, then the multiplicity of T as a simple constituent of

↓G

HP (S) is equal to the multiplicity of S as a simple constituent of↑ G HP (T ).

By Clifford theory for Mackey algebras the restriction of a simple μR(G)-module to a

sub-normal subgroup is semisimple [13]. In the next result we give an explicit description of the restriction of simple Mackey functors of p-groups over characteristic p.

Corollary 3.5. Suppose that G is a p-group. Then for any K G L H ,

(i) ↓G_LP_K,G_F LgK⊆G P_LL_∩g_K,_F. (ii) ↓GLSH,GF LgNG(H )⊆G:gHL SgL_H,_F.

Proof. (i) By 3.1(ii) P_K,G_F ↑G_KB_FK, and using the Mackey decomposition formula we get

↓G LP G K,F ↓ G L↑ G KB K F LgK⊆G ↑L L_∩g_KBL∩ g_K F .

And by 3.1(ii) each↑L_L_∩g_KBL∩ g_K

F is isomorphic to PLL∩g_K,_F, implying the result.

(ii) For each XLLlet mXbe the multiplicity of SX,LFas a simple constituent of↓GLSGH,F.

Then by 3.4(iii) mX is the multiplicity of P_H,G_Fas a direct summand of↑G_LP_X,L_F. Moreover by

3.2(ii) we have↑G_LP_X,L_F P_X,G_F. Therefore mX= 1 if H =GX, and mX= 0 otherwise. Since

↓G LS

G

H,Fis semisimple we must have

↓G LS G H,F XLL: X=GH S_X,L_F.

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4. Relative projectivity

The object of this section is to provide some applications of Green’s indecomposability the-orem for Mackey algebras 2.4 and the results of the previous section. Specifically, we obtain Mackey algebra versions of some classical results about dimensions of modules of group alge-bras, and we derive some results related to restriction and induction of Mackey functors.

A Mackey functor M for G overK is said to be H -projective for some subgroup H of G if M is a direct summand of↑G_H↓G_HM, equivalently M is a direct summand of↑G_HT for some Mackey functor T for H . For an indecomposable Mackey functor M, up to conjugacy there is a unique minimal subgroup H of G, called the vertex of M, so that M is H -projective. Although these notions are very similar to the corresponding notions in group algebras, there are some crucial differences. For example, projective Mackey functors may not be 1-projective. Indeed, over any field the projective indecomposable Mackey functor P_H,VG has vertex H , see [9,12]. These differences are major obstacles to the obtaining Mackey algebra versions of the classical results about group algebras. That is why we usually need some extra assumptions for Mackey algebras to satisfy the similar results.

Let V be a principal indecomposableFG-module. Then dim_FV is divisible by the order of a Sylow p-subgroup of G. This is one of the earliest result in modular representation theory, known as Dickson’s theorem, see [7, Corollary 7.16, p. 91]. We begin by obtaining a Mackey algebra version of Dickson’s theorem.

Proposition 4.1. Let P be a p-subgroup of G. Then, given any 1-projective indecomposable

μ_F(G)-module M , there is a positive integer n such that↓G_PM nμ_F(P )t₁1as μ_F(P )-modules. In particular dim_Fμ_F(P )t₁1divides dim_F↓G_PM .

Proof. We may write↓G_PM T1⊕ · · · ⊕ Tn as a direct sum of indecomposable μF(P )

-mod-ules Ti. Then each Ti is a direct summand of ↓GPM, and as M is 1-projective by applying the

Mackey decomposition formula we see that each Ti is 1-projective. Therefore Ti is a direct

summand of↑P₁T for some indecomposable μ_F(1)-module T . As μ_F(1) F, T F and Ti is

a direct summand of↑P₁T μ_F(P )t₁1⊗_FF μ_F(P )t₁1. Moreover by 3.1 the μ_F(P )-module

μ_F(P )t₁1is indecomposable, implying that each Ti is isomorphic to μF(P )t11. 2

Let V be a finitely generated P -projectiveFG-module where P is a p-subgroup of G. Then |S : P | divides dimFV where S is a Sylow p-subgroup of G containing P , see [8, Theorem 7.5, p. 293]. We next give a Mackey algebra version of this result.

Proposition 4.2. Suppose M is a finitely generated P -projective μ_F(G)-module where P is a normal p-subgroup of G. If S is a Sylow p-subgroup of G, then|S : P | divides dim_F↓G_PM .

Proof. Evidently P ⊆ S. As M is P -projective it is a direct summand of ↑G_P↓G_PM. Using the Mackey decomposition formula we see that↓G_SMis a direct summand of

↓G

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We write↓G_PM T1⊕ · · · ⊕ Tr as a direct sum of indecomposable μF(P )-modules Ti. By 2.4

each↑S_PTi is an indecomposable μF(S)-module. Therefore

↓G

SM n1↑SPT1⊕ · · · ⊕ nr↑SPTr

for some integers ni 0. Consequently,

↓G PM ↓ S P↓ G SM n1↓SP↑ S PT1⊕ · · · ⊕ nr↓SP↑ S PTr.

Finally, using the Mackey decomposition formula we see that ↓S P↑ S PTi gP⊆S g_T i

for each i. In particular, for any i,|S : P | divides dim_F↓S_P↑S_PTi, implying the result. 2

For group algebras the dimension of ↓G_H↑G_HW is divisible by |G : H | which may not be the case for Mackey algebras unless H is normal. This is one of the reasons why we assumed the normality of P in the previous result. In fact, let G be a p-group having a nonnormal subgroup H of order p. Then for any g∈ G it is clear that H ∩gH = H if and only if g∈ NG(H ), and H∩gH= 1 otherwise. By using the Mackey decomposition formula we see

that dim_F↓G_H↑G_HS_H,H_F= |NG(H ): H | is not divisible by |G : H|. In particular, the normality

of P cannot be removed from the statement of 4.2 so as to leave a correct statement.

Corollary 4.3. Let G have a normal Sylow p-subgroup S, and let M be a finitely generated

indecomposable μ_F(G)-module withdim_F↓G_SM= 1. Then S is a vertex of M.

Proof. As S is a Sylow p-subgroup,F is of characteristic p and ↓G_SM= 0 it follows that M

is S-projective, see [12]. Let Q be a vertex of M and Q be a vertex of↓G_SM. Then Q⊆ S, and if Q= S then we choose a maximal subgroup P of S that contains Q. Then↓G_SM is a finitely generated P -projective μ_F(S)-module and P is normal in S. Therefore 4.2 implies that |S : P | = p divides dimF↓SP↓GSM= dimF↓GPM which is impossible because↓GPM⊆ ↓GSM.

Therefore S is a vertex of↓G_SM. As Q is a vertex of M it is a direct summand of↑G_Q↓G_QM. This implies that↓G_SMis a direct summand of↓_SG↑G_Q↓G_QMwhich is, by the Mackey decomposition formula, isomorphic to_gS_⊆G↑Sg_Q↓Gg_QM. Moreover as dimF↓G_SM= 1, the module ↓G_SMand

the modules↓Gg_QMmust be indecomposable. Finally by 2.4 we see that↓G_SM ↑Sg_Q↓Gg_QMfor

some g∈ G, in particular ↓G_SM isgQ-projective. As S is a vertex of↓G_SM we conclude that

S= Q. 2

Remark 4.4. Let G have a psubgroup H . Then, any finitely generated indecomposable H -projective μ_F(G)-module is projective.

Proof. Let M be such a module. Then M is a direct summand of↑G_HT for some finitely gen-erated indecomposable μ_F(H )-module T . As H is a p-group, μ_F(H ) is semisimple [11], implying that T is projective. Lastly, as restriction and induction are exact functors which are

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two sided adjoints of each other, they send projectives to projectives [11,12]. Hence M must be projective. 2

5. A Swan length theorem

Suppose N is a normal subgroup of G and G/N is a p-solvable group. Let U be a simple FG-module. Then, the composition length of ↓G

NU divides|G : N|. This is a theorem of Swan,

see [10] or [7, Theorem 9.20, p. 143]. Our aim is to show that a similar result holds for Mackey algebras. We first need the following, see [13].

Theorem 5.1 (Clifford Theorem for Mackey algebras). Let N be a normal subgroup of G and

S_H,VG be a simple μ_K(G)-module with H N. Then, there is a simple μ_K(N )-submodule SN_H,W of↓G_NS_H,VG such that ↓G NS G H,V d gL⊆G |g NS N H,W and ↓ NG(H ) NN(H )V d gT⊆NG(H ) cg_H(W )

for some positive integer d, called the ramification index of S_H,VG relative to N , where L and T are the respective inertia groups of S_H,WN and W in G and NG(H ). Furthermore, L= NT and

NL(H )= T .

Theorem 5.2 (Swan Length Theorem for Mackey algebras). Let N be a normal subgroup of G

such that G/N is p-solvable, and let M be a simple μ_F(G)-module with↓G_NM= 0. Then the composition length of↓G_NM divides|G : N|.

Proof. Let M= S_H,VG where H N. Then by 5.1 ↓G NS G H,V d gL⊆G |g NS N H,W and ↓ NG(H ) NN(H )V d gT⊆NG(H ) c_Hg(W )

where W is a simple NN(H )-module which is a direct summand of↓NG(H )

NN(H )V, and L and T are

the respective inertia groups of S_H,WN and W . Moreover L= NT and NL(H )= T . Now

NG(H )/NN(H )= NG(H )/ NG(H )∩ N NG(H )N /N G/N

implies that NG(H )/NN(H )is p-solvable. Thus by [7, Theorem 9.20, p. 143] the composition

length of↓NG(H )

NN(H )V, which is d|NG(H ): T |, divides |NG(H ): NN(H )|. So there is a positive

integer s such that sd|NG(H ): T | = |NG(H ): NN(H )| implying that

sd=T : NN(H ) =T : N ∩ NL(H ) =|T : N ∩ T | = |NT : N| = |L : N|.

Hence sd|G : L| = |G : N|, and so the composition length of ↓G

NM, which is d|G : L|, divides

|G : N|. 2

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Corollary 5.3. Let N be a normal subgroup of G such that G/N is p-solvable, and let M be a

simple μ_F(G)-module with↓G_NM= 0. If S is a simple μ_F(N )-module which is a direct summand of↓G_NM , thendim_F↓G_NM divides|G : N| dim_FS.

Suppose A is a normal abelian subgroup of G so that G/A is p-solvable. Then dim_FUdivides |G : A| for any simple FG-module U, see [3,10]. As a consequence of the previous result we have the following similar result for Mackey algebras.

Corollary 5.4. Let N be a normal abelian p-subgroup of G such that G/N is p-solvable, and

let M be a simple μ_F(G)-module with↓G_NM= 0. Then dim_F↓G_NM divides|G : N|.

Proof. As N is abelian, 2.3(iii) implies that every simple μ_F(N )-module is one dimensional. Then the result follows. 2

The proof of the above result depend on the fact that for an abelian p-group G, all simple

μ_F(G)-modules are one dimensional. It can be seen easily that all simple μ_K(G)-modules are one dimensional if and only ifK is of characteristic p > 0 and G is a p-group such that all subgroups of G are normal.

6. A Fong dimension theorem

For any natural number n we let npand np denote its p- and p-part, respectively. We denote

by P ( ) the projective cover of its argument.

If G is a p-solvable group and V is a simpleFG-module then dim_FP (V )= |G|p(dimFV )p.

This is known as Fong’s dimension theorem, see [7, Theorem 16.9, p. 230]. In this section we obtain a result which looks like Fong dimension theorem for Mackey algebras.

We now give some results related to ramification indices and restriction of principal indecom-posable Mackey functors. The next two results will be the main ingredients of the proof of 6.3, a result which we suggest as a Fong dimension theorem for Mackey algebras.

Proposition 6.1. Let N be a normal subgroup of G such that G/N is a p-group and M be a

simple μ_F(G)-module with↓G_NM= 0. Then

(i) P (M) ↑G_NP (S) for any simple μ_F(N )-module S which is a direct summand of↓G_NM .

(ii) The ramification index of M relative to N is 1.

(iii) For any simple μ_F(N )-module S which is a direct summand of↓G_NM ,

↓G NP (M) gN⊆G |g NP (S) |L : N| gL⊆G |g NP (S)

where L is the inertia group of S. In particular,↓G_NP (M) |L : N|P (↓G_NM).

Proof. Take any simple μ_F(N )-module S which is a direct summand of ↓G_NM. Then by 5.1 ↓G

NM d

gL⊆G| g

NS where d is the ramification index. Using 3.4(iii) we see that d is the

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Hence d= 1 and P (M) ↑G_NP (S). Finally, observing that L is also the inertia group of P (S), the last part follows by the Mackey decomposition formula applied to the first part. 2

Under the assumptions of 6.1, if S is a simple μ_F(N )-module then P (↑G_NS) ↑G_NP (S). In-deed, as induction is an exact functor sending projectives to projectives it follows that P (↑G

NS)is

a direct summand of↑G_NP (S)from which the isomorphism is concluded by 2.4. Moreover, any primitive idempotent of μ_F(N )stays primitive in μ_F(G), because given a primitive idempotent

eof μ_F(N )we have μ_F(G)e ↑G_Nμ_F(N )ewhich is indecomposable by 2.4. More to the point, any finitely generated N -projective indecomposable μF(G)-module M is of the form↑G_NT for some indecomposable μ_F(N )-module T .

Proposition 6.2. Let N be a normal subgroup of G such that G/N is a p-group, and M be a simple μ_F(G)-module with↓G_NM= 0. Then

(i) For any simple μ_F(N )-module S which is a direct summand of↓G_NM ,

↓G NP (M) d gL⊆G |g NP (S)

where d is the ramification index of M relative to N and L is the inertia group of S in G. In particular,↓G_NP (M) P (↓G_NM).

(ii) p does not divide d.

Proof. (i) Let S be any simple μ_F(N )-module which is a direct summand of↓G_NM. It follows by 5.1 that↓G_NM d_gL_⊆G|g_NSwhere d is the ramification index and L is the inertia group of S. Let S1, . . . , Sr be a complete set of representatives of simple μF(N )-modules. By the functorial

properties of restriction it sends projective modules to projective modules (if the resulting module is nonzero), see [11,12]. So we may write

↓G NP (M) r i=1 niP (Si)

for some integers ni 0. Note that P (Si) P (Sj)if and only if i = j. Then by 3.4 ni is

the multiplicity of M as a simple constituent of↑G_NSi. Therefore ni is equal to the dimension of

Homμ_F(G)(P (M),↑G_NSi). Moreover, as G/N is a p-group, [13, Corollary 3.8] implies that↑G_NSi

is semisimple. Now using the adjointness of restriction and induction we have asF-modules Homμ_F(G) P (M),↑G_NSi Homμ_F(G) M,↑G_NSi Homμ_F(N ) ↓G NM, Si d gL⊆G Homμ_F(N ) |g NS, Si .

Hence ni= d if Si |g_NSfor some g∈ G, and ni= 0 otherwise, implying the result.

(ii) Let M= S_H,VG where H N. Then by 5.1 we can choose S as S = S_H,WN where W is a simpleFNN(H )-submodule of V . By 5.1 we know that the ramification index d of M relative to

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to a subgroup of G/N , the group NG(H )/NN(H )is a p-group. So by the corresponding result

in the context of group algebras [8, p. 389], p does not divide d. 2

Theorem 6.3 (Fong Dimension Theorem for Mackey algebras). Let G be a p-solvable group,

N be a normal subgroup of G, and let M be a simple μ_F(G)-module with↓G_NM= 0.

(i) If N is a p-group then

dim_F↓G_NP (M)= n dim_F↓G_NM for some natural number n which is a power of p.

(ii) If N is a p-group then

dim_F↓G_NP (M)= |G/N|p

dim_F↓G_NM_pdim_FP (S)

where S is a simple μ_F(N )-module which is a direct summand of ↓G_NM . And if X is a minimal subgroup of M then dim_FP (S)= dim_F(μ_F(N )t_XX).

Proof. As G/N is p-solvable we may find a chain N= Nr⊂ Nr−1⊂ · · · ⊂ N1= G where each

Ni is a normal subgroup of G, and each quotient Ni/Ni+1has order ni which is a p-number or

a power of p (consider the pp-series of G/N ).

Let S1= M and for i = 2, . . . , r choose a simple μF(Ni)-module Si which is a direct

sum-mand of↓Ni−1

Ni Si−1, and let di−1be the ramification index of Si−1relative to Ni, and let Li be

the inertia group of Siin Ni−1. So in particular Ni ⊆ Li⊆ Ni−1.

Then, for i= 2, . . . , r it follows by 6.1 and 6.2 that ↓Ni−1 Ni Si−1 di−1 giLi⊆Ni−1 |gi NiSi and ↓ Ni−1 Ni P (Si−1) ki−1 giLi⊆Ni−1 |gi NiP (Si)

which is a p-number, if ni₋₁= |Ni₋₁: Ni| is a p-number.

Then, as each Niis normal in G, we have

↓G NM= ↓ Nr−1 Nr · · · ↓ N2 N3↓ N1 N2S1 (d1d2· · · dr−1) g2L2⊆N1 g3L3⊆N2 · · · grLr⊆Nr−1 |g2g3···gr Nr Sr, ↓G NP (M)= ↓ Nr−1 Nr · · · ↓ N2 N3↓ N1 N2S1 (k1k2· · · kr−1) g2L2⊆N1 g3L3⊆N2 · · · grLr⊆Nr−1 |g2g3···gr Nr P (Sr). Therefore dim_F↓G_NM= _r₋₁ i=1 di _r₋₁ i=1 |Ni: Li+1| dim_FSr, dim_F↓G_NP (M)= _r₋₁ i=1 ki _r₋₁ i=1 |Ni: Li+1| dim_FP (Sr).

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(i) Suppose N is a p-group, then μ_F(N )is semisimple [11]. So P (Sr)= Sr. Letting n be the

numberr_i₌₁−1(ki/di)we see that n is a power of p and dimF↓G_NP (M)= n dimF↓G_NM.

(ii) Suppose N is a p-group. Let X be a minimal subgroup of M. By 5.1 we may take Sr to be

S_X,N_F, whose dimension is equal to|N : NN(X)| by 2.3. Then 3.1 implies that P (Sr) μF(N )tXX.

Let{1, 2, . . . , r −1} = I J where i ∈ I if and only if |Ni: Ni+1| = niis a power of p. So, di= 1

i∈I|Ni: Ni+1| = |G/N|p, the result follows. 2

Let us explain how to derive, using the previous result, Fong’s dimension theorem for group algebras. Given a simple FG-module V , we use 6.3(ii) with M = S_1,VG and N = 1 to get dim_F↓G₁P (M)= |G|p(dimF↓G1M)pdimF(μF(1)t11), where, of course, dimF(μF(1)t11)= 1 and

(dim_F↓G₁M)p = (dimFV )p. Moreover dimF↓G1P (M)= dimFP1,VG (1)= dimFP (V ), because

by [12, 12.6] we have P_1,VG (1) P (V ). Hence dim_FP (V )= |G|p(dimFV )p.

7. Mackey functors of direct products

We give some results about Mackey functors for direct products of groups. If G1and G2are

groups thenK(G1× G2) KG1⊗KKG2, and if we assume that G1 and G2 have coprime

orders, then it follows by 2.1 that μ_K(G1× G2) μK(G1)⊗KμK(G2), because subgroups of

G1×G2are of the form H1×H2for some subgroups Hiof Gi. We first recall some basic notions

about tensor products of algebras, see [2, pp. 249–254]. Let A and B be finite dimensional K-algebras and K be algebraically closed. If X and Y are A- and B-modules, respectively, then

X⊗_KY becomes an A⊗_KB-module by means of the action (a⊗ b)(x ⊗ y) = ax ⊗ by. The simple modules of A⊗_KBare, up to isomorphism, precisely the modules X⊗_KY where X and

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X1, X2and indecomposable B-modules Y1, Y2 then Xi⊗KYi is an indecomposable A⊗KB

-module for i= 1, 2, and X1⊗KY1 X2⊗KY2if and only if X1 X2and Y1 Y2. Let X be

an A-module and Y be a B-module. Identifying A and B with their images A⊗ 1 and 1 ⊗ B in

A⊗_KBwe see that X⊗_KY is isomorphic to (dim_KY )Xand (dim_KX)Yas A- and B-modules, respectively. In particular, if X⊗_KY is a projective A⊗_KB-module then X and Y are projective

A- and B-modules.

Proposition 7.1. Let G1and G2be groups with coprime orders and let H2be a subgroup of G2.

Suppose thatK is algebraically closed. Then, any finitely generated 1 × H2-projective

indecom-posable μ_K(G1× G2)-module is of the form S⊗KT , as μK(G1)⊗KμK(G2)-module, where

S is a principal indecomposable μ_K(G1)-module and T is a finitely generated H2-projective

indecomposable μ_K(G2)-module. Conversely, given such modules S and T the μK(G1× G2

)-module S⊗_KT is finitely generated,1× H2-projective and indecomposable.

Proof. Let M be such a μ_K(G1× G2)-module. Then there is a finitely generated 1× H2

-projective indecomposable μ_K(1×G2)-module T such that M is a direct summand of↑G1_×G21×G2T,

and there is a finitely generated indecomposable μ_K(1× H2)-module U such that T is a direct

summand of↑1₁×G2_×H2U. As ↑1×G2

1×H2U μK(1× G2)⊗μK(1×H2)U μ_K(G2)⊗μ_K(H2)U,

T may be regarded as a finitely generated H2-projective indecomposable μK(G2)-module.

Writ-ing μ_K(G1) S1⊕ · · · ⊕ Snas a direct sum of principal indecomposable μK(G1)-modules we

note that ↑G1×G2 1×G2 T μK(G1× G2)⊗μK(1×G2)T μ_K(G1)⊗KμK(G2) ⊗μ_K(G2)T μK(G1)⊗_K μ_K(G2)⊗μ_K(G2)T μK(G1)⊗_KT n i=1 Si⊗KT .

Since each Si⊗KT is indecomposable, M must be of the desired form.

Conversely, let S and T be given. As T is H2-projective we may regard T as a 1× H2

-projective μ_K(1× G2)-module. Evidently, S⊗KT is a direct summand of μK(G1)⊗KT which

is equivalent, by what we have done above, to↑G₁_×G21×G2T. Since T is 1× H2-projective the result

follows. 2

We next provide some applications of 7.1. For a prime number p and a group G, we denote by Op(G)and Op(G)the respective largest normal p and normal p-subgroups of G.

Corollary 7.2. Let G be a nilpotent group and H be a p-subgroup of G. Then the dimension of

the μ_F(G)-module P_H,VG is divisible by the dimension of μ_F(Op(G))t_HH.

Proof. Applying 7.1 with G1= Op(G) and G2= Op(G) we see that P_H,VG S ⊗FT for

some projective indecomposable μ_F(Op(G))-module T with vertex H . Then T is isomorphic to

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Let M be a μ_K(G)-module with vertex H . Then there is an indecomposable μ_K(H )-module

Usuch that M is a direct summand of↑G_HU. Any such U is called an H -source of M.

Corollary 7.3. Let G be a nilpotent group and H be a p-subgroup of G. If M is a finitely

generated indecomposable μ_F(G)-module with vertex H and H -source U , thendim_F↑O_Hp(G)U divides dimFM .

Proof. By the proof of 7.1 there is a finitely generated indecomposable μ_F(Op(G))-module

T with vertex H and H -source U such that M S ⊗_FT for some principal indecomposable

μ_F(Op(G))-module S. In particular, dimFT divides dimFM. As T is a direct summand of

↑Op(G)

H Uit follows by 2.4 that T ↑ Op(G)

H U, finishing the proof. 2

Finally we give a result on the simple μ_F(G)-modules for a nilpotent group G.

Corollary 7.4. Let G be a nilpotent group. Then for a simple μ_F(G)-module SG_H,V we have

dim_FS_H,VG = |G : NG(Op(H ))| dimFS O_p(G) O_p(H ),V.

Proof. Considering SG_H,V as a simple μ_F(Op(G))⊗FμF(Op(G))-module, there are simple

modules M1= S

Op(G)

X,W and M2= S Op(G)

Y,F for some respective subgroups X and Y of Op(G)

and Op(G) such that SH,VG M1⊗FM2. As H is isomorphic to Op(H )× Op(H ) we see

that M1(Op(H )) and M2(Op(H )) are nonzero. On the other hand, since M1(X)and M2(Y )

are nonzero it follows that S_H,VG (XY ) is nonzero. Hence we may take X= Op(H )and Y =

Op(H ). Moreover as SH,VG (H )= V we have V M1(Op(H ))⊗FM2(Op(H )) W ⊗FF as

FOp(H )⊗FFOp(H )-modules. So W V . Finally by 2.3 dimFM2= |G : NG(Op(H ))|. 2

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